The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  X  X  X  X  1  1  1  1  1  X  1  1  X  X  1 X^2  1  1  X  2  1 X^2  1  1  X  0  1  1  X X^2 X^2  0 X^2 X^2
 0 X^2+2  0 X^2  0  0 X^2 X^2+2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  0  2  0 X^2+2 X^2 X^2 X^2  2 X^2 X^2+2  0  2  0 X^2+2 X^2+2 X^2+2 X^2+2 X^2  0  0 X^2  2 X^2  0 X^2  2  2  2  0 X^2+2 X^2  0 X^2+2  2 X^2  2  0
 0  0 X^2+2 X^2  2 X^2 X^2+2  2  2 X^2 X^2+2  2  0 X^2+2 X^2  0  0 X^2 X^2+2  0  2 X^2 X^2+2 X^2 X^2  0  2 X^2+2 X^2 X^2+2 X^2 X^2+2 X^2 X^2+2 X^2+2 X^2+2  2  2 X^2+2 X^2 X^2 X^2 X^2+2  0 X^2  2  0 X^2+2  2  2  2 X^2 X^2

generates a code of length 53 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+191x^52+55x^56+5x^60+4x^64

The gray image is a code over GF(2) with n=424, k=8 and d=208.
This code was found by Heurico 1.16 in 24.2 seconds.